Confirmatory Factor Analysis
Exploratory factor analysis can be used to identify common factors and factor structure among a set of observed variables / indicators. Confirmatory factor analysis (CFA) can be used to study how well a hypothesized factor model fits a new sample from the same population or a sample from a different population. The CFA model is the same as the EFA model with the exception that restrictions can be placed on factor loadings, variances, covariances, and residual variances resulting in a more parsimonious model. Using CFA, one can
- investigate if a factor model fits a new sample from the same population – the confirmatory aspect.
- evaluate if a factor model fits a sample from a different population – measurement invariance,
- study the behavior of new measurement items embedded in a previously studied measurement instrument, and
- estimate factor scores.
An example
The study by Karl J. Holzinger and Frances Swineford involves data collection from two schools, the Grant-White School (\(n=145\)) and Pasteur School (\(n=156\)). In the EFA example, we have identified 4 factors with data from the the Grant-White School. We now investigate whether the same factor model would fit the data in the Pasteur School. The data are saved in the file Pasteur.csv.
Confirmatory factor analysis
The model
The path diagram for the model is given in the figure below. Note that we have assumed there are 4 factors. And the indicators of each factor are also known. This model is based on the EFA of Grant-White school data with the factor loadings greater than 0.3 kept in the model.
Model estimation
To estimate a confirmatory factor model, the R package lavaan can used. A confirmatory factor model cannot be identified without proper constraints, that's, to fix some parameters to be known values in the model. The reason is that factors are unmeasured and thus have no scales. To identify a model, the factors have to be given specific scales. There are two ways to do this. First, we can fix the variance of a factor to be 1. This is to standardize the factor. Second, we can fix the loading of one observed variable or indicator to be one. This is essentially to make the factor to have the same scale as the observed variables. Fixing a factor loading or a factor variance is the minimum requirement to identify a factor model. However, this does not guarantee a model is identifiable. Practically, if a model cannot be identified, the software used to estimate the model will not run correctly.
The cfa() function in lavaan can be used to estimate a factor model. To use the function, we need to first specify the factor model. A factor model take the format
factor =~ y1 + y2 +y3
Note that we use the symbol "=~" to define a factor. The factor is on the left of the symbol and the indicators are on the right of it.
The R code for the example is given below.
> library(lavaan)
This is lavaan 0.5-23.1097
lavaan is BETA software! Please report any bugs.
> usedata('Pasteur.csv')
>
> cfa.model<-'
+ spatial =~ visual+cubes+paper+lozenge+straight+figurer
+ verbal =~ general+paragrap+sentence+wordc+wordm
+ speed =~ add+code+counting+straight
+ memory =~ wordr+numberr+figurer+object+numberf+figurew
+ '
>
> cfa.est<-cfa(cfa.model, data=Pasteur)
> summary(cfa.est, fit=TRUE)
lavaan (0.5-23.1097) converged normally after 251 iterations
Number of observations 156
Estimator ML
Minimum Function Test Statistic 238.751
Degrees of freedom 144
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 1149.278
Degrees of freedom 171
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.903
Tucker-Lewis Index (TLI) 0.885
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -9899.892
Loglikelihood unrestricted model (H1) -9780.517
Number of free parameters 46
Akaike (AIC) 19891.785
Bayesian (BIC) 20032.078
Sample-size adjusted Bayesian (BIC) 19886.474
Root Mean Square Error of Approximation:
RMSEA 0.065
90 Percent Confidence Interval 0.050 0.079
P-value RMSEA <= 0.05 0.050
Standardized Root Mean Square Residual:
SRMR 0.079
Parameter Estimates:
Information Expected
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
spatial =~
visual 1.000
cubes 0.396 0.089 4.465 0.000
paper 0.233 0.051 4.547 0.000
lozenge 1.087 0.182 5.961 0.000
straight 1.599 0.623 2.567 0.010
figurer 0.535 0.139 3.836 0.000
verbal =~
general 1.000
paragrap 0.282 0.024 11.846 0.000
sentence 0.463 0.035 13.322 0.000
wordc 0.388 0.038 10.177 0.000
wordm 0.589 0.047 12.585 0.000
speed =~
add 1.000
code 0.835 0.145 5.758 0.000
counting 0.708 0.147 4.811 0.000
straight 1.091 0.265 4.115 0.000
memory =~
wordr 1.000
numberr 0.537 0.106 5.040 0.000
figurer 0.486 0.106 4.570 0.000
object 0.379 0.070 5.372 0.000
numberf 0.277 0.060 4.638 0.000
figurew 0.245 0.055 4.444 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
spatial ~~
verbal 21.927 5.770 3.800 0.000
speed 24.531 9.382 2.615 0.009
memory 10.542 4.998 2.109 0.035
verbal ~~
speed 66.093 17.349 3.810 0.000
memory 10.027 7.727 1.298 0.194
speed ~~
memory 47.001 14.950 3.144 0.002
Variances:
Estimate Std.Err z-value P(>|z|)
.visual 21.376 4.533 4.716 0.000
.cubes 19.547 2.401 8.139 0.000
.paper 6.472 0.799 8.102 0.000
.lozenge 52.062 7.715 6.748 0.000
.straight 875.004 110.732 7.902 0.000
.figurer 40.094 5.568 7.200 0.000
.general 41.308 5.979 6.909 0.000
.paragrap 4.185 0.571 7.330 0.000
.sentence 6.615 1.058 6.252 0.000
.wordc 13.215 1.664 7.943 0.000
.wordm 14.218 2.063 6.892 0.000
.add 415.730 55.840 7.445 0.000
.code 75.538 19.745 3.826 0.000
.counting 280.324 35.795 7.831 0.000
.wordr 83.348 12.917 6.452 0.000
.numberr 43.595 5.790 7.529 0.000
.object 16.578 2.329 7.119 0.000
.numberf 15.587 1.982 7.865 0.000
.figurew 14.000 1.752 7.989 0.000
spatial 28.852 6.417 4.496 0.000
verbal 96.582 15.354 6.290 0.000
speed 200.381 59.608 3.362 0.001
memory 60.814 15.974 3.807 0.000
>
Output
By default, the cfa() function fixes a factor loading for each factor to be 1 and estimates the rest factor loadings. Different from EFA, CFA also test the significance of the factor loadings based on a z-test. For example, the factor loading for cubes on factor 1 is 0.396 with the standard error 0.089. The corresponding z-value is 4.465 with a p-value almost 0. Therefore, this factor loading is statistically significant from 0. In addition, it also estimates the factor variances and covariances as well as the uniqueness factor variances.
Model fit evaluation
CFA provides a lot more information to evaluate whether a model fits a sample. First, a chi-square test can be used. For this test, the null hypothesis is that the model fits the data well. Therefore, one would hope to get a small chi-square statistic and a corresponding large p-value. Typically, when p-value is larger than 0.5, one fails to reject the model.
There are many other fit statistics and indices that can be used to evaluate model fit. The most widely used ones include CFI, RMSEA, and SRMR.
- Comparative fit index (CFI) compares the model under evaluation with a baseline model. One form of the baseline model is the independent model where it assumes the independence among the observed variables. In general, such a baseline model would not fit the data well. Therefore, CFI measures the relative improvement of the current model. It is generally accepted that a CFI greater than 0.96 (or 0.95) indicates a good fit model.
- Root mean square error of approximation (RMSEA) is a measure of chi-square attributed to each participate after controlling the model complexity. Therefore, a smaller value indicates better fit. In the literature,
- a RMSEA $\leq 0.05$ indicates a model fits the data closely.
- a $0.05<$ RMSEA $\leq 0.08$ indicates a model fits the data reasonably well.
- a RMSEA $>0.1$ indicates a model is a bad model.
- Standardized root mean square residual (SRMR) measures the difference between the observed covariance matrix from the data and the predicted covariance matrix based on the model. A value SRMR=0 indicates a perfect of the model. In general, a value less than 0.08 is considered a good fit of the model under evaluation.
Using these criteria, we can evaluate whether the confirmatory factor model identified using the data from Grant-White school fits the data from the Pasteur school well.
- The chi-square statistic (
Minimum Function Test Statisticin the output) is 238.75 with the degrees of freedom 144 and a p-value close to 0. Therefore, one would reject the hypothesis that the model fits the data simply based on it. - Comparative Fit Index (CFI) is 0.903, which is smaller than the cut-off value 0.95. It also suggests a bad fit.
- The RMSEA = 0.065, which lies the range of a reasonable fit model.
- The SRMR = 0.079, which is smaller than but close to the cut-off value 0.08.
Overall, the chi-square test and the other criteria suggest the model barely fits the data. Therefore, we cannot replicate the factor structure identified in EFA for the Grant-White school data in the Pasteur school.
Estimate the model with standardized factors
Instead of fitting a factor loading to be 1, we can also fix the factor variances to be 1. The R code below does the analysis. Note that the model fit is exactly the same as before. However, we can now directly estimate the factor correlation matrix.
> library(lavaan)
This is lavaan 0.5-23.1097
lavaan is BETA software! Please report any bugs.
> usedata('Pasteur.csv')
>
> cfa.model<-'
+ spatial =~ visual+cubes+paper+lozenge+straight+figurer
+ verbal =~ general+paragrap+sentence+wordc+wordm
+ speed =~ add+code+counting+straight
+ memory =~ wordr+numberr+figurer+object+numberf+figurew
+ '
>
> cfa.est<-cfa(cfa.model, data=Pasteur, std.lv=TRUE)
> summary(cfa.est, fit=TRUE)
lavaan (0.5-23.1097) converged normally after 191 iterations
Number of observations 156
Estimator ML
Minimum Function Test Statistic 238.751
Degrees of freedom 144
P-value (Chi-square) 0.000
Model test baseline model:
Minimum Function Test Statistic 1149.278
Degrees of freedom 171
P-value 0.000
User model versus baseline model:
Comparative Fit Index (CFI) 0.903
Tucker-Lewis Index (TLI) 0.885
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -9899.892
Loglikelihood unrestricted model (H1) -9780.517
Number of free parameters 46
Akaike (AIC) 19891.785
Bayesian (BIC) 20032.078
Sample-size adjusted Bayesian (BIC) 19886.474
Root Mean Square Error of Approximation:
RMSEA 0.065
90 Percent Confidence Interval 0.050 0.079
P-value RMSEA <= 0.05 0.050
Standardized Root Mean Square Residual:
SRMR 0.079
Parameter Estimates:
Information Expected
Standard Errors Standard
Latent Variables:
Estimate Std.Err z-value P(>|z|)
spatial =~
visual 5.371 0.597 8.993 0.000
cubes 2.124 0.433 4.901 0.000
paper 1.254 0.250 5.012 0.000
lozenge 5.837 0.790 7.385 0.000
straight 8.589 3.240 2.651 0.008
figurer 2.872 0.697 4.121 0.000
verbal =~
general 9.828 0.781 12.580 0.000
paragrap 2.773 0.234 11.850 0.000
sentence 4.551 0.340 13.389 0.000
wordc 3.811 0.375 10.172 0.000
wordm 5.790 0.459 12.605 0.000
speed =~
add 14.156 2.105 6.723 0.000
code 11.823 1.224 9.658 0.000
counting 10.024 1.673 5.990 0.000
straight 15.440 3.235 4.773 0.000
memory =~
wordr 7.798 1.024 7.614 0.000
numberr 4.185 0.683 6.129 0.000
figurer 3.790 0.707 5.363 0.000
object 2.953 0.434 6.798 0.000
numberf 2.160 0.398 5.428 0.000
figurew 1.913 0.374 5.121 0.000
Covariances:
Estimate Std.Err z-value P(>|z|)
spatial ~~
verbal 0.415 0.085 4.895 0.000
speed 0.323 0.103 3.129 0.002
memory 0.252 0.109 2.303 0.021
verbal ~~
speed 0.475 0.081 5.902 0.000
memory 0.131 0.098 1.336 0.181
speed ~~
memory 0.426 0.097 4.410 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.visual 21.376 4.533 4.716 0.000
.cubes 19.547 2.401 8.139 0.000
.paper 6.472 0.799 8.102 0.000
.lozenge 52.062 7.715 6.748 0.000
.straight 875.006 110.732 7.902 0.000
.figurer 40.093 5.568 7.200 0.000
.general 41.308 5.979 6.909 0.000
.paragrap 4.185 0.571 7.330 0.000
.sentence 6.615 1.058 6.252 0.000
.wordc 13.215 1.664 7.943 0.000
.wordm 14.218 2.063 6.892 0.000
.add 415.731 55.841 7.445 0.000
.code 75.538 19.745 3.826 0.000
.counting 280.324 35.795 7.831 0.000
.wordr 83.347 12.917 6.452 0.000
.numberr 43.595 5.790 7.529 0.000
.object 16.578 2.329 7.119 0.000
.numberf 15.587 1.982 7.865 0.000
.figurew 14.000 1.752 7.989 0.000
spatial 1.000
verbal 1.000
speed 1.000
memory 1.000
>
To cite the book, use:
Zhang, Z. & Wang, L. (2017-2026). Advanced statistics using R. Granger, IN: ISDSA Press. https://doi.org/10.35566/advstats. ISBN: 978-1-946728-01-2.
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